The invention relates to an assembly of accelerometers for application in a system for measuring the three dimensional movements of a rigid body which accelerometers are connected to said rigid body in a spacial configuration with reference to an imaginary orthogonal coordinate system.
In general ballistometry is the collection of techniques and algorithms used for the reconstruction of the motion of a free-floating body from a record of measurements. This includes satellite attitude reconstruction from onboard measurements but also the tracking of motions of a rigid body by any of a variety of methods. These methods may include the use of optical sensors, gyroscopes, accelerometers or cinematographic or photogrammetric observation of position an attitude of a rigid body.
There are several reasons to opt for the use of accelerometers, one of which is that accelerometers and their associated electronics require the least mass and volume. These factors weigh heavily in space research programs, one of the possible application fields of this invention.
The motion of a rigid body can always be analyzed into a linear and an angular velocity. The kinematic variables appear as parameters in the representation of the acceleration field that is related to the rigid body. The number of accelerometers used, their location on the rigid body and the relative positioning of their sensitive axes in the acceleration field determine the resulting number and value of the kinematic parameters as well as the computations necessary for the reconstruction of the motion of the rigid body.
Prior art ballistometry focuses on the number of accelerometers and the relative positioning of their sensitive axes. In this respect it is common knowledge that a minimum of six linear accelerometers is required for a complete definition of the kinematic variables of a rigid body. Five linear accelerometers are required to compute all three components of angular acceleration about the body-fixed axes of the acceleration field of the rigid body. A sixth accelerometer is needed to provide, in addition, all three components of linear acceleration for complete definition of rigid body motion.
The computation of angular acceleration of a rigid body from measured linear accelerations is a relatively simple procedure based on well-known kinematic principles. The determination of arbitrary motion with six sensors involves (numerical) integration or differentiation. Because of errors in measurement, these stepwise integration or differentiation procedures usually result in an accumulation of errors. These problems are described by PADGOANKER et al in "Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers." (in Journal of Applied Mechanics, September 1975, pages 552-556). To solve these problems Padgoanker et.al. introduce the use of nine accelerometers instead of six, position them in a predetermined spacial configuration with their sensitive axes directed such that by relatively simple calculations on the accelerometer outputs the linear and angular acceleration components of the motion of the rigid body to which the accelerometers are attached can be determined. More specifically the prior art configuration comprises nine accelerometers, three of which are located at the origin of an imaginary orthogonal coordinate system which is fixed with respect to the rigid body of which the motions are measured. The sensitive axis of these three accelerometers are trained respectively in the direction of Z, Y and X-axis of the coordinate system. This prior art nine accelerometer configuration furthermore comprises a set of two accelerometers on each of the orthogonal axes of the coordinate system at a predetermined distance from the origin. The sensitive axes of the two accelerometers positioned on the X-axis of the coordinate system are trained respectively parallel to Y-axis and parallel to the Z-axis. The sensitive axes of the two accelerometers positioned on the Y-axis of the coordinate system are trained respectively parallel to the X-axis and parallel to the Z-axis. The sensitive axes of the two accelerometers positioned on the Z-axis of the coordinate system are trained respectively parallel to the X-axis and parallel to the Y-axis.
Rotation of the rigid body can cause problems when the rotation is three-dimensional and there are errors in the measured linear accelerations. As stated before errors in measurement can result in an accumulation of errors when stepwise (numerical) computations are performed. MITAL et.al. introduce in "Computation of Rigid-Body Rotation in Three-Dimensional Space From Body-Fixed Linear Acceleration Measurements." (in Journal of Applied Mechanics, Vol. 46, December 1979, page 925-930) a method which generates an orthogonal transformation matrix, which needs to be evaluated only when it is required to transform a position vector from the body-fixed frame to the inertially fixed reference frame.
It will be clear now that a nine sensor arrangement can allow a direct determination of linear acceleration (a) of the origin O of the acceleration field of the rigid body as well as the angular velocity (.omega.) and of the angular acceleration (.omega.) by algebraic operations on the accelerometer output. This is a stable calculation and leaves scope for additional extraction of parameter values from the comparison of angular velocity and angular acceleration by calculation and measurement.
However the nine sensor arrangement according to prior art has three accelerometers at the origin of an imaginary orthogonal coordinate system which is fixed with respect to the rigid body of which the motions have to be measured. For no motion of the center of mass of the rigid body it is clear that in that case the origin of the coordinate system should be put at the center of mass of the rigid body and measurement of angular velocity is possible with three sensors. Usually a free-floating rigid body will show only small excursions in the center of mass location and it is therefore advantageous to have the origin O of the coordinate system near the approximate center of mass position. However in general this location is centrally located in the rigid body to which the sensors are attached and is inaccessible. These circumstances make it desirable to have an arrangement that has no sensors at the origin of the coordinate system. None of the arrangements in the quoted literature fulfil this requirement.